Spatial GMRF Q Model INLA Extensions References Gaussian Markov Random Fields Johan Lindstr ¨ om 1 1 Centre for Mathematical Sciences Lund University Pan-American Advanced Study Institute B´ uzios June 18, 2014 Johan Lindstr¨ om - [email protected]Gaussian Markov Random Fields 1/33
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Spatial GMRF Q Model INLA Extensions References
Gaussian Markov Random Fields
Johan Lindstrom1
1Centre for Mathematical SciencesLund University
Pan-American Advanced Study InstituteBuzios
June 18, 2014
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Spatial GMRF Q Model INLA Extensions References Interpolation Big N Alternatives
Spatial interpolation — Kriging
Given observations at some locations, Y(si), i = 1 . . .nwe want to make statements about the value at unobservedlocation(s), X(s).
In the simplest case we assume a Gaussian model for the data
[YX
]∼ N
([μy
μx
],
[Σyy Σyx
Σ⊤yx Σxx
]),
with some parametric form for the covariance matrix and mean
Y ∼ N (μ(θ),Σ(θ)) .
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Spatial GMRF Q Model INLA Extensions References Interpolation Big N Alternatives
The “Big N” problem
The log-likelihood becomes
l (θ|Y) = −1
2log |Σ(θ)| −
1
2
(Y − μ(θ)
)⊤
Σ(θ)−1(
Y − μ(θ)).
Given (estimated) parameters, predictions at the unobservedlocations are given by
E(
X∣∣∣Y, θ
)= μx +ΣxyΣ
−1yy (Y − μy).
The “Big N” problem
Given N observations:
◮ The covariance matrix has O(N2
)unique elements.
◮ Computations scale as O(N3
)(due to |Σ| and Σ−1).
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Spatial GMRF Q Model INLA Extensions References Interpolation Big N Alternatives
Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
Global Temperature Data
January 2003 July 2003
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
Global Temperature Data
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
GMRFs based on SPDEs (Lindgren et al., 2011)
GMRF representations of SPDEs can be constructed forto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(κ2 −Δ)(τ x(s)) = W(s), s ∈ Rd
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
GMRFs based on SPDEs (Lindgren et al., 2011)
GMRF representations of SPDEs can be constructed forto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(κ2 −Δ)(τ x(s)) = W(s), s ∈ Ω
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
GMRFs based on SPDEs (Lindgren et al., 2011)
GMRF representations of SPDEs can be constructed forto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(κ2 eipθ −Δ)(τ x(s)) = W(s), s ∈ Ω
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
GMRFs based on SPDEs (Lindgren et al., 2011)
GMRF representations of SPDEs can be constructed forto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(κ2s +∇ · ms −∇ · Ms∇)(τsx(s)) = W(s), s ∈ Ω
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
GMRFs based on SPDEs (Lindgren et al., 2011)
GMRF representations of SPDEs can be constructed forto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(∂∂t
+ κ2s,t +∇ · ms,t −∇ · Ms,t∇
)(τs,tx(s, t)) = E(s, t), (s, t) ∈ Ω × R
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Spatial GMRF Q Model INLA Extensions References Global Data General SPDEs
Questions?
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Spatial GMRF Q Model INLA Extensions References
Bibliography I
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Bibliography II
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Bibliography III
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Simpson, D., Lindgren, F., and Rue, H. (2010), “In order to make spatialstatistics computationally feasible, we need to forget about the
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Bibliography IV
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Stein, M. L., Chi, Z., and Welty, L. J. (2004), “Approximating likelihoods
for large spatial data sets,” J. R. Stat. Soc. B, 66, 275–296.
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fields,” Comput. Stat. Data Anal., 56, 49–61.
Whittle, P. (1954), “On Stationary Processes in the Plane,” Biometrika, 41,434–449.
— (1963), “Stochastic processes in several dimensions,” Bull. Int. Stat.
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